It would "probably" be fine. If you experience crashes or weird behavior, then definitely look at upgrading PSU, but you wouldn't need to go crazy on it.

You would get MUCH better performance from a 7770, and they have one at $95 on Newegg. $75 after rebate. Not sure with that CPU, though. There is a point where you won't see better performance because you're limited by CPU. I've only been following the Intel chips. Gave up on AMD a few years ago.

You PC has a 300W PSU of questionable quality most likely, but like frameunit said, it might be sufficient. Speaking of frameunit's suggestion, there's a R5 250X (rebranded 7770) that is 60$ after rebate. Anyway, here are mine: